How many hidden layers are needed for this training data set
You need to perform Hyperparameter Tuning to identify -


Number of hidden layers.
Number of neurons in each of the hidden layers.
Dropout
The activation function you use in each of your hidden layers.


There parameters are only related to how you build your model. There are others that relate to training like batch size, number of epochs and so on. Your model's performance ultimately depends on how well you tune your hyperparameters.

Also note that hyperparameter tuning is a trial and error task because it depends on several factors that may not be obvious to us. With experience, experts do build certain thumb rules about what may be the right choice, but there is no way to generalize it. "Some say the number of hidden layers is roughly the number of separation times" - is just another thumb rule. You simply need to find out what best suits your scenario.